Course Archives Theoretical Statistics and Mathematics Unit

Course: Measure Theory Level: Postgraduate Time: Currently not offered

Syllabus Past Exams Syllabus: i) The concept of s-algebra, Borel subsets of R, Construction of Lebesgue and Lebesgue-Stieltjes measures on the real line following outer measure. ii) Abstract measure theory: definition and examples of measure space, measur- able functions, Lebesgue integration, convergence theorems (Fatous Lemma, Monotone convergence and dominated convergence theorem). iii) Caratheodory extension theorem, completion of measure spaces. iv) Product measures and Fubinis theorem. v) Lp-spaces, Riesz-Fischer Theorem, approximation by step functions and con- tinuous functions. vi) Absolute continuity, Hahn-Jordan decomposition, Radon-Nikodym theorem, Lebesgue decomposition theorem. Functions of bounded variation. vii) Complex measures. If time permits: Vitali covering lemma, differentiation and fundamental theorem of calculus. Suggested Texts : (a) H. L. Royden and Patrick Fitzpatrick: Real Analysis, Pearson, 4th edition. (b) Robert B. Ash and Catherine A. Doleans-Dade, Probability and measure theory, GTM(211), Academic Press, 2nd edition. (c) Elias M. Stein, Rami Shakarchi, Real Analysis: Measure Theory, Integration and Hilbert Spaces , Princeton Lectures in Analysis. (d) Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics, A Wiley Series. (e) G. de Barra, Measure Theory and Integration. Top of the page Past Exams

Midterm 24.pdf 26.pdf Semestral 22.pdf 24.pdf 26.pdf Supplementary and Back Paper 26.pdf Top of the page

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